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Continuous Closed Form Trajectories Generation and
Control of Redundantly Actuated Parallel Kinematic
Manipulators
Moussab Bennehar∗ , Ahmed Chemori∗ , Sbastien Krut∗ and Franois Pierrot∗
∗ Robotics
Department, LIRMM, Univ. Montpellier 2 - CNRS
161 rue Ada, 34090 Montpellier, France.
Email: bennehar(chemori, krut, pierrot)@lirmm.fr
Abstract—This paper deals with continuous closed form trajectories generation and control of redundantly actuated parallel
kinematic manipulators. Continuous closed form trajectories are
usually given by means of analytical functions of time and show
inherent discontinuities in velocity and/or acceleration profiles. In
order to solve this problem, we propose an approach consisting
of modifying the time function without changing the overall
closed shape of the trajectory in the operational space. To deal
with the actuation redundancy, an extended version of the PD
controller with computed feedforward is proposed. The resulting
control torques are projected using a regularization matrix in
order to remove internal forces that may be dangerous for the
manipulator. The proposed trajectory generator as well as the
controller are implemented in real-time on Dual-V; a redundantly
actuated parallel manipulator developed in our laboratory.
I.
I NTRODUCTION
Parallel Kinematic Manipulators (PKMs) are mostly known
for their better performances over their serial counterparts
[9]. Indeed, in contrast with the serial ones, the actuators of
PKMs are located on the fixed base and, hence, result in a
very lightweight moving mass [3]. Furthermore, the closed
kinematic chains yield to a stiffer and more accurate structure.
Nevertheless, PKMs exhibits many drawbacks that moderate
their expansion in industrial applications. The abundance of
singularities and the limited workspace are the most noteworthy ones. While the latter is a matter of optimal design,
the former can be solved by actuation redundancy. Hence,
Redundantly Actuated PKMs (RA-PKMs) are increasingly
attracting more interest of researchers in the last two decades.
Actuation redundancy significantly improves the performance of PKMs by homogenizing the physical properties
throughout the workspace, allowing the manipulator to achieve
higher accelerations. However, despite the aforementioned
qualities, PKMs potential is not yet completely explored and is
still an open research field. From one hand, mechanical design,
identification and optimization can be further investigated in
order to increase their acceptance in industrial applications.
From the other hand, the close-loop constraints and the inherent nonlinearities give rise to more challenging tasks in terms
of trajectory generation and control.
This research was supported by the French National Research Agency,
within the project ANR-ARROW
Thanks to their high dynamic performances, PKMs are
typically used for industrial high-speed applications such as
pick-and-place in food industry [10]. Hence, most researches
in terms of trajectory generation focused only on classical
Point-to-Point (PtP) trajectories [5], [4], [11]. Nevertheless,
the inherent stiffness and accuracy of PKMs awarded them
to be used in more complex industrial tasks such as laser
cutting [12] or machining [13]. In this case, Continuous Closed
Shape (CCS) trajectories draw more attention than traditional
PtP ones. This class of trajectories, however, has not been
sufficiently investigated in the literature.
It is conventional that PKMs share many physical properties with serial robots. Consequently, most of the control
literature developed for serial manipulators has been straightforwardly implemented on PKMs [14], [15]. However, in the
case of RA-PKMs, a particular property characterizing this
class of manipulators needs a special attention. The nonuniqueness of the inverse dynamics solution yields to control
forces with no effects on the moving platform of the robot [6].
These control forces produce undesired internal pre-stress in
the kinematic chains that can harm the manipulator and cause
a loss of energy. Consequently, the control scheme should take
into consideration such a phenomenon.
In this work, we address the two aforementioned problems,
namely; the CCS trajectories generation and the reduction
of internal forces for RA-PKMs. We propose a trajectory
generator that takes into consideration continuity constraints,
both in task and joint spaces. As regards to the control solution,
a joint space PD with computed feedforward controller [16]
is proposed. This control scheme has the advantage of being
lightweight in terms of computation time while compensating
some of the nonlinearities. In order to deal with the internal
forces issue, the proposed controller is extended with a projection of the generated control torques with the regularization.
The proposed approach is then validated in real-time on a 3
dof RA-PKM, namely, the Dual-V.
This paper is organized as follows. In section 2, the
dynamic model of the Dual-V robot is presented. Section 3
is devoted to the proposed trajectory generator. The proposed
control solution is detailed in Section 4. Experimental results
are presented in Section 5. Finally, conclusions and perspectives are drawn in Section 6.
II.
DYNAMIC MODELING OF THE D UAL -V ROBOT
Dual-V is a 3 dof planar redundantly actuated parallel
manipulator belonging to the 4-RRR family. The arrangement
of the four chains allows three independent movements for
its end-effector (a traveling plate): two translations in the
plane and one rotation about the z-axis. Hence, the Cartesian
coordinates the traveling plate are described by the vector
X = [x, y, θ ]T . For the dynamic modeling of the robot, the
approach developed in [3] has been extended to take into
account the rotational inertia of the forearms.
Crank
Coupler
Traveling plate
Length [m]
0.2800
0.2800
0.22
Mass [Kg]
1.169
0.606
0.899
Inertia [Kgm2 ]
0.012967
0.006417
0.008168
TABLE I: Dual-V geometric and dynamic parameters
The required torque to move the robot’s mechanical
structure can be decomposed into three sub-torques, namely;
τ1 , τ2 , τ3 ∈ R4 . Each sub-torque is responsible for the movement of a part of the structure. The torque τ1 is the required
torque to move the traveling plate and a part of the mass of
the forearms, it can be calculated using the equations of power
of the actuators as follows:
Fig. 1: The CAD of the Dual-V with parameters definition
τ1 = JmT ∗ MI X¨
(1)
where MI ∈ R3×3 is the mass matrix of the concerned part of
∗
the links to be moved in Cartesian space, JmT = Jm (JmT Jm )−1 ∈
4×3
R
is the pseudo-inverse of the transpose of the inverse
Jacobian matrix Jm and X¨ ∈ R3 is the acceleration vector of
the traveling plate. The component τ2 is the required torque
to move the arms, the counter-masses and the remaining part
of the couplers, it can be expressed as:
¨
τ2 = MII q¨ = MII (J˙m X˙ + Jm X)
(2)
where MII ∈ R4×4 is the mass matrix regrouping dynamic
parameters of the involved parts of the structure in joint space,
X˙ ∈ R4 is the traveling plate velocity vector and J˙m is the time
derivative of the inverse Jacobian matrix. Up to now, only the
inertia of the two mass model [8] of the couplers is considered.
Though this is a righteous hypotheses when light material is
used, it fails when the links are made with heavy material such
as steal or aluminum. This is the case of the Dual-V robot and,
hence, an additional term should be added. The torque term
τ3 is the additional torque component to compensate for the
real inertia of the couplers and that of the equivalent mass
model. Due to the number of pages limitation, the details of
this additional term are not provided herein and the reader
is referred to [8] for a full description of the Dual-V robot
˙ X)
¨
dynamic modeling. It is worth noting though that τ3 (X, X,
is a highly nonlinear function with respect to its arguments.
Finally, the full dynamic model of the Dual-v is written by
regrouping the three sub-torques as follows:
¨ + τ3 (X, X,
˙ X)
¨
τ = JmT ∗ MI X¨ + MII (J˙m X˙ + Jm X)
III.
R EFERENCE TRAJECTORIES GENERATION
CCS trajectories are more complex than traditional PtP
ones [4],[5] (e.g pick-and-place trajectories) and many of them
are generated by means of analytical functions parametrized
by the time variable t. Therefore, generating trajectories with
continuity constraints on velocities and accelerations may not
be possible using their original analytical equations. Continuity
of trajectories is a crucial constraint for the robot drives since
discontinuities in the trajectories can generate discontinuous
control torques and thus, lead to undesired behavior of the
mechanical structure such as vibrations or instabilities.
In this paper we are interested in the class of trajectories
described by a sum of weighted sin and cosine functions.
This class of trajectories has been chosen because it covers a
large amount of simple and complex geometrical shapes (e.g.
circles, ellipses, deltoids, . . . ). The general analytical form of
the trajectories in question is given by:
n n n
2i
1i
x(t) = ∑ ai cosαi 2π t + bi sinβi 2π t
T
T
i=1
(4)
where T ∈ R+ is the trajectory duration and ai , αi , bi , βi , n ji ∈
R; i = 1, . . . , n; j = 1, 2 are the parameters defining the shape
of the trajectory.
In the sequel, we will first give an illustrative example
of direct application of (4). Then, its major drawback is
highlighted and a solution to overcome it is proposed.
A. Classical method
(3)
The CAD model of the Dual-V with parameters definition
is depicted in Fig. 1 and its geometric and dynamic parameters
are summarized in Table I. Further details about the Dual-V
robot are given in [7] and [2].
Let xd (t) be the desired Cartesian position defined by (4)
to be tracked by the traveling plate of the robot. These desired
trajectories are fully included in the workspace of the robot
away from singularities. The desired velocity and acceleration
trajectories for this motion are obtained by differentiating xd (t)
with respect to time, thus:
(5)
1.5
1
1
y−position [mm]
n 2π
1i
ai αi n1i sinαi −1 2π t −
vd (t) = −
∑
T i=1
T
n 2i
bi βi n2i cosβi −1 2π t
T
Cartesian positions [mm]
1.5
n
0.5
0
−0.5
−1
vd (0) = −
(6)
2
2π
T
n
∑ bi βi n2i = f (bi , βi , n2i )
(7)
∑ ai αi (αi − 1)n21i = f (ai , αi , n1i )
(8)
i=1
n
i=1
therefore, the resulting trajectories can be velocity and/or
acceleration discontinuous leading to discontinuous control
torques and therefore big tracking errors or even instability.
To further illustrate this inconvenience let’s consider a
simple circular trajectory with a radius r and a center whose
coordinates are (xc , yc ) = (0, 0). The resulting trajectory needs
to be C2 continuous (in velocities and accelerations) both
in operational and joint spaces in order to be appropriately
tracked by the traveling plate. The analytical equations of
positions, velocities and accelerations are given by:
0.5
xd (t) = r cos
yd (t) = r sin
2π
T (t − t0 )
2π
T (t − t0 )
2
0
−5
−10
0
x˙ d
y˙ d
0.5
1
Time [s]
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x−position [mm]
These trajectories satisfy equations (4), (5) and (6) being
only shifted in time by t0 . Fig. 2 illustrates the plots of the
obtained trajectories using (9), (10) and (11) (t0 = 0.5s, T =
1s). One can notice the presence of discontinuities on both
velocity and acceleration profiles that have to be removed. In
what follows, we propose a novel method to overcome this
drawback and generate C2 continuous reference trajectories.
20
0
−20
−40
−60
0
x
¨d
y¨d
0.5
1
Time [s]
1.5
2
In order to overcome the discontinuity problem in the
velocity and acceleration trajectories, we propose to revisit the
general form of the desired operational trajectories. Consider
again the previous illustrative example of a circular trajectory.
If we check the previous equations of the trajectory, it can be
seen that they can be rewritten as follows:
xd (t) = r cos(λ (t))
yd (t) = r sin(λ (t))
(12)
With λ (t) = 2π
T (t −t0 ) which is a simple affine function of
time t and when implemented, does not satisfy any continuity
constraint on the velocity and acceleration trajectories. The
corresponding velocities and accelerations are obtained by
differentiating (12) with respect to time, which gives:
x˙d (t) = −r λ˙ (t) sin(λ (t))
y˙d (t) = r λ˙ (t) cos(λ (t))

h
i
 x¨d (t) = −r λ¨ (t)sin(λ (t)) + λ˙ 2 (t)cos(λ (t))
h
i
 y¨d (t) = r λ¨ (t)cos(λ (t)) − λ˙ 2 (t)sin(λ (t))
(10)
(11)
2
40
B. Proposed solution: continuous trajectories with a velocity
trapezoidal profile
(9)
1.5
60
Fig. 2: View of the generated Cartesian trajectories using the
classical method
2π
T (t − t0 )
2π
T (t − t0 )
1.5
5
2π
x˙d (t) = −r 2π
T sin T (t − t0)
2π
y˙d (t) = r 2π
T cos T (t − t0 )
2
x¨d (t) = −r( 2π
T ) cos
2π 2
y¨d (t) = −r( T ) sin
1
Time [s]
10
Without loss of generality, assume that the robot starts
and finishes its movement with a null velocity and a null
acceleration (i.e. v(0) = v(T ) = 0 and a(0) = a(T ) = 0). Using
the original analytical functions (5) and (6), however, may
lead to non-zero initial velocity and/or acceleration. Indeed,
if we replace t = 0 or t = T in (5) and (6) we will obtain
values dependent on initial and final conditions and thus, does
not necessarily equal zero. For instance, the obtained initial
conditions are given by:
2π
ad (0) = −
T
−1.5
n
n 1i
2
αi −2
2π t
a
α
(α
−
1)n
cos
i
i
i
∑
1i
T
i=1
n 2i
+ bi βi (βi − 1)n22i sinβi −2 2π t
T
xd
yd
Cartesian accelerations [mm/s2]
2
Cartesian velocties [mm/s]
2π
ad (t) = −
T
0
−0.5
−1
−1.5
0
0.5
(13)
(14)
From (13) and (14) one can notice that if λ (t) is chosen
such that it satisfies the following boundary constraints:
λ (t0 ) = 0; λ˙ (t0 ) = 0; λ¨ (t0 ) = 0
λ (t0 + T ) = 2π; λ˙ (t0 + T ) = 0; λ¨ (t0 + T ) = 0
(15)
one can ensure the continuity on the velocity and acceleration trajectories. One possible solution would then be to
choose λ (t) with a trapezoidal profile on velocity. Indeed,
this choice allows to specify boundary conditions and hence,
8
1.5
6
˙
λ(t)
4
2
4
2
Original
Proposed
0
0.5
1
Time [s]
0
0.5
1.5
1
Time [s]
Cartesian velocties [mm/s]
40
¨
λ(t)
20
0
−20
−40
Original
Proposed
1
Time [s]
1.5
the constraints (15) can be satisfied. A comparison between
trapezoidal and linear velocity profile is depicted in Fig. 3.
The analytical equations of λ (t) are now expressed as:
λ f −λ0
[−t 4 + 2τt 3 ] + λ0 ; t0 ≤ t ≤ τ
2τ 3 (T −τ)
λ f −λ0
τ
τ ≤ t ≤ T −τ
(T −τ) [t − 2 ] + λ0 ;
λ f −λ0
4
[(t − T ) + 2τ(t − T )3 ] + λ f ;
2τ 3 (T −τ)
(16)
else
where τ is the duration of initial acceleration and final deceleration phases that has to be chosen according to the robot
dynamic capabilities. λ0 and λ f are the initial and final values
respectively for λ (t). For instance, in the case of the above
circular trajectory example, λ0 = 0 and λ f = 2π.
The obtained trajectories using the proposed strategy are
now continuous in velocity and acceleration and result in the
same trajectory in operational space as the original one (cf. Fig.
4). Therefore, they can be safely used in the control scheme
without risks on the actuators.
IV.
−1
P ROPOSED CONTROL SOLUTION
It has been shown that PKMs exhibit many similarities
with their serial counterparts [1]. Consequently, many control
strategies that were mainly developed for serial manipulators
have been successfully applied on PKMs. However, in the
particular case of RA-PKMs, we notice the existence of
antagonistic control torques not affecting the motion of the
manipulator. Thus, using conventional control strategies will
certainly give occurrence to internal forces (incompatible with
the robot’s kinematics) [6] and hence, create internal prestress in the mechanism. The antagonistic forces can cause
a multitude of undesirable phenomena such as the loss of
energy, instability and vibrations. Consequently, the control
architecture has to take in consideration this peculiarity.
A. Joint space PD controller
Consider a RA-PKM with m degrees of freedom and n > m
actuators and let the joints position vector be denoted by q ∈
0
−0.5
−1.5
0.5
1
Time [s]
1.5
2
5
0
−5
−10
0.5
−1
xd
yd
10
−15
0
Fig. 3: Evaluation of λ (t), λ˙ (t) and λ¨ (t): original (dashed
line) and modified (solid line)


λ (t) =


λ (t) =


 λ (t) =
0
−0.5
15
60
−60
0.5
1
0.5
−1.5
0
1.5
1.5
1
−2
x˙ d
y˙ d
0.5
1
Time [s]
1.5
2
−1.5
−1
−0.5
0
0.5
1
1.5
2
x−position [mm]
Cartesian accelerations [mm/s2]
λ(t)
6
Cartesian positions [mm]
Original
Proposed
y−position [mm]
8
100
50
0
−50
−100
0
x
¨d
y¨d
0.5
1
Time [s]
1.5
2
Fig. 4: Obtained C2 continuous operational trajectories
Rn . The PD control is a decentralized uncoordinated control
strategy relying on the measured error between the desired
and the actual position. Let e(t) be the joints error vector,
i.e : e(t) = qd (t) − q(t), where qd (t) denotes the desired joint
trajectory. Then, the PD control action is expressed by:
τPD = K p e(t) + Kd e(t)
˙
(17)
where K p and Kd are positive definite feedback gain matrices
usually chosen diagonal. If the actuators are all identical, which
is usually the case for PKMs, the same feedback gains could be
selected for each axis, namely; K p = k p In×n and Kd = kd In×n ,
where In×n is an identity matrix and k p and kd are positive
scalars denoting the feedback gains that should be tuned to
achieve satisfactory closed-loop performance. Good accuracy
on high accelerations is mostly required in parallel robots.
B. Desired compensation by nonlinear feedforward
If the inverse dynamics of the robots are well known,
it would be interesting to exploit this knowledge to further
improve the tracking performance. Indeed, the tracking errors
can be significantly improved by compensating the inherent
nonlinearities. One possible way would be the addition of a
feedforward term. This strategy enables in partially compensating the nonlinear dynamics in the feedforward way, i.e. only
desired values of positions, velocities and accelerations are
fed to the inverse dynamics model. For the case of the DualV robot, the inverse dynamics are given by (3), hence, the
feedforward torque is computed as:
τ f f = JmT ∗ MI X¨d + MII (J˙m X˙d + Jm X¨d ) + τ3 (Xd , X˙d , X¨d )
(18)
where the subscript d refers to the desired quantities.
C. Projection method to reduce internal forces
For RA-PKM, the control torques may contain antagonistic
forces that does not create any motion as they are in the
null-space of JmT and, thus, need to be removed. This can be
achieved by projecting the input torques onto the range of JmT
by the following projector [6]:
Xd , X˙d , X¨d Inverse
Dynamics
Kd
d
dt
τf f
Xd
Inverse
Kinematics
qd
+
e
−
Kp
+ τPD +
+
+
τ
∗
JmT JmT
τ∗
q
RA-PKM
q
Fig. 5: Block diagram of the proposed control scheme
equations. In order to further investigate the benefits of the
proposed approach, four different geometric shapes are implemented, namely; a circle, an ellipse, a tear-like and a deltoide
(cf. Fig. 7a). The traveling plate of the manipulator has to
perform every geometric trajectory in T = 0.5 s. Between the
end point of a shape trajectory and the starting point of the next
one, the end-effector has to perform a PtP trajectory defined
by a 5th order polynomial function.
The controller feedback gains have been experimentally
tuned using the trial-and-error technique. Usually, industrial
robots are not equipped with velocity sensors, thus velocities
have to be estimated. In our case, a carefully designed low-pass
filter is used to estimate the velocities.
Fig. 6: View of the experimental setup; 1 the parallel manipulator, 2 the host PC, 3 the target PC, 4 emergency
stop, 5 host monitor, 6 xPC target monitor
RJmT = I − NJmT
(19)
where NJmT = I − JmT ∗ JmT is the projector to the null-space of
JmT . Hence, the internal antagonistic forces are eliminated by
projecting the control torques onto the range space of JmT . This
is achieved by using the projection matrix RJmT as:
τ ∗ = RJmT (τPD + τ f f )
(20)
The block diagram of the proposed control scheme is
depicted in Fig 5.
V.
R EAL - TIME EXPERIMENTAL RESULTS
A. The Dual-V robot and its hardware configuration
The Dual-V links are all made with Aluminum and
mounted on four direct drive actuators provided by ETEL
Motion Technology. The actuators are fixed on an Aluminum
base and can supply up to 127 N.m torques. We use Simulink
and Real-Time Workshop (both from Mathworks Inc.) to
implement and build the control algorithm. The generated
code is then uploaded to the target PC; an industrial computer
cadenced at 10 kHz and running xPC Target in real-time. The
experimental setup is depicted in Fig. 6.
B. Real-time experimental results
The proposed trajectory generator as well as the proposed
controller presented in the previous section were implemented
on the Dual-V robot. For comparison purpose, we also implement the classical approach using the original unmodified
Figures 7b and 7c depict the Cartesian tracking performance of the end-effector on both x and y axes. We can clearly
see that in the case of original trajectories, the discontinuities
cause a poor tracking performance. This is more noticeable
at the beginning and the end of the circle and the ellipse.
If the robot was equipped with accelerometers, one could
notice the high frequency vibrations. However, if we check
the obtained performance using the proposed approach, we
can clearly see the improvements in terms of tracking errors at
the end and starting points. It can be clearly seen that the proposed trajectories are accurately tracked thanks to the removed
discontinuities. The improvements are more noticeable on the
starting and end points of the circle and ellipse trajectories
(t = 5, 5.5, 7.5 and 8s). These results were expected since the
main purpose of this paper is to present a technique to remove
these discontinuities from the trajectories generated by means
of classical analytical functions. These discontinuities are a
major source of tracking loss.
The control inputs generated when using the proposed
method as well as those generated by classical one are depicted in Fig. 8. As expected, the controller generates very
high torques when it faces discontinuities on the reference
trajectories, because the drives need to generate high torques
in order step from zero to a certain velocity in a very short
time, which is practically very delicate.
VI.
C ONCLUSIONS AND FUTURE WORK
In this paper we investigated the continuous closed shape
trajectories generation and the reduction of internal forces by
means of control for RA-PKMs. These trajectories are defined
using analytical functions that show inherent discontinuities.
For this reason, we proposed in this work an approach to
overcome this inconvenience for a specific class of trajectories
by modifying the time function without changing the overall
closed shape of the trajectory in operational space. Furthermore, to overcome the phenomenon of internal forces, a PD
with computed feedforward controller has been implemented.
The resulting control torques have been projected using the
Jacobian matrix to reduce internal forces that can harm the
mechanical structure. The latter helps in significantly reducing
the energy consumption and the vibrations on the robot’s
traveling plate. To validate the proposed approaches, realtime experiments have been conducted on a real 3 dof planar
redundantly actuated parallel manipulator developed in our
laboratory. The proposed trajectory generator exhibited its
superiority over the original trajectory generator that utilizes
original analytical equations.
0.4
5
0
−5
−10
−15
0.2
0
−0.2
−0.4
−0.6
−20
−10
0
10
x−position [mm]
y−axis error [mm]
x−axis error [mm]
y−position [mm]
10
20
(a) Reference trajectory
1
Original
Proposed
15
6
8
10
Time [s]
0.5
0
−0.5
−1
12
Original
Proposed
(b) Error on x axis
6
8
10
Time [s]
12
(c) Error on y axis
15
15
10
10
10
5
5
5
5
0
0
0
−5
−5
−5
−5
−10
−10
−10
−10
−15
6
8
10
Time [s]
−15
12
6
8
10
Time [s]
−15
12
6
8
10
Time [s]
−15
12
15
15
10
10
10
5
5
5
5
0
0
τ4 [N.m]
15
10
τ3 [N.m]
15
τ2 [N.m]
τ1 [N.m]
0
τ4 [N.m]
15
10
τ3 [N.m]
15
τ2 [N.m]
τ1 [N.m]
Fig. 7: Reference trajectory and evaluation of the Cartesian tracking errors versus time
0
−5
−5
−5
−10
−10
−10
−10
−15
−15
−15
8
10
Time [s]
12
6
8
10
Time [s]
12
6
8
10
Time [s]
12
8
10
Time [s]
12
6
8
10
Time [s]
12
0
−5
6
6
−15
Fig. 8: Evolution of control torques: (top) original discontinuous trajectories and (bottom) proposed continuous trajectories
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[2]
[3]
[4]
Hui Cheng, Yiu, Y.-K., and Zexiang Li., Dynamics and Control of Redundantly Actuated Parallel Manipulators, Transactions on Mechatronics,
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Wijk, V. van der, Krut, S., Pierrot, F., and Herder, J. L., Generic Method
for Deriving the General Shaking Force Balance Conditions of Parallel
Manipulators with Application to a Redundant Planar 4-RRR Parallel
Manipulator, The 13th World Congress in Mechanism and Machine
Science, 2011.
[9]
[10]
[11]
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